The Limit Theorems for Extreme Residuals in Nonlinear Regression Model with Gaussian Stationary Noise

Abstract

In this paper non-linear regression model with Gaussian stationary random noise and continuous time is considered. The behavior of normalized in some way maximum residuals and maximum of residuals absolute values in which its the least squares estimator is substituted instead of unknown parameter of regression function. The convergence of distribution of these normalized maximum to double exponent law is proved which follows from the assumption of random noise normality. In the normalization of this maximum instead of unknown variance and the 2nd spectral moment of Gaussian stationary random noise consistent estimates of these parameters are substituted. It generalizes the residuals sum of squares of the classical regression analysis and Lindgren’s the 2nd spectral moment estimator, accordingly. In the paper mathematical machinery of statistics of random processes and limit theorems for extremes of Gaussian stationary noise is used. The obtained results can be used in construction of statistical tests for adequacy of the regression model.